PROPORTION. 



PROPORTION. 



ua oolouriese or slightly yellow-coloured liquid of agreeable odour, 

 ineulubl* in tr. but soluble in alcohol or ether. 1U bulling point U 

 313* Khr. It U iulUmmablr, burning with a pale blue llauic. Kin ic 

 aid converts it into propionic acid. 



Prepionic auliyJriJr, or ankyilrviu propionic <""'' ({'{}*,)'}) ' 



colourless liquid formed on distilling a mixture of two equivalent* of 



ride .i | li. -i h^niH, and six of dried propionate of soda. It 



has a disagreeable odour, U not soluble in water, and boils at 829* Fahr. 



Pnpiouie arid ( g<J } ) Uttocttic acid, or *iftaatoic acid. 



ThU acid U one of the product* of the oxidation of oleic ncid ; it la 

 also produced by the action of strong cauiti* potash upon sugar, 

 ktarch, gum, or mannit<>, and by the action of ferments upon glycerin. 

 The moat convenient method of preparing it is to add cyanide of 

 ethyl drop by drop to a concentrated alcoholic solution of potash 

 contained in a retort. The reaction that then takes place is thus 

 expressed: 



C.U., C,X + HO + KO, HO = KO, O.H.O, -f NH, 



Cyanide of ethyl. Water. Potash. 



PropionaU of 

 pota*h. 



Ammonia. 



On dUtilling the residue with sulphuric acid, or with syrupy phos- 

 phoric acid, pure propionio acid passes over. 



Propionic acid crystallines in plate*. On the application of a gentle 

 heat it futes, and at 284 Fabr. boils. It is very sour, and has a 

 pungent odour, somewhat like a mixture of butyric and acrylic acids. 

 Water take* up a large quantity of it, excess floating like an oil upon 

 the saturated solution. 



fnijiivnatet, or the salU of propionic acid, are mostly crystalline, 

 and soluble in water. They are all monometallic, and easily decom- 

 posed by strong acids. Propionatt of potatk is very deliquescent. 

 Pnifiionatc of ttxla crystallises with, difficulty ; it forms a double salt 

 with acetate of soda, crystallising in needles, and containing (C,H,Na 

 p., C,H,Nab. + 9 Aq). Propionatt of baryta (C.H.BaO.) crystallises 

 in prUms. Propionate of lime forms silky fibres. Pmpiimate of copper 

 (C.H.t'uO, + Aq.) crystallises in small oblique prisms: it is very 

 soluble in alcohol, but only slightly so in water. Pragumait of lead is 

 uncrystalluable. Propionatt of rilver (C,H,AgO,) crystallises in 

 tufts of needles ; it forms a davllt salt with acetate of silver, contain- 

 ing (C,H,AgO 4 , C,H s AgO,). Propinnate of ttlnjl ; ^n^i'mic tlhcr ; 

 tutacelic tthrr (CJljiC.HjiO,) is obtained on distilling propionate 

 of soda with alcohol and sulphuric acid. It U lighter than water, 

 boils at 213 8 Fahr., and has an odour resembling rum. 



yitropropiofic acid (C H,(N0 4 )0,), nitromttatetic and, is a heavy 

 fellow oil slightly soluble in water, but very soluble in alcohol. It 

 poaaoMn an aromatic odour and a sweet taste, and is intlaii: 

 burning with a reddish flame. It forms crystalline monometallic salts 

 with base*. For the method of preparing it, see BUTTBIO ACIP, 

 lutyronr. 



/C.H.O, 



Propionamide, 01 nitride ofpropionj/l > N I results from the 



/C.H. 

 l [ 

 \ H 



of ammonia on propionate of ethyl. 

 OPORTION. There must be in 



action 



PROPORTION. There must be in the mind of every person, 

 antecedently to all mathematical instruction, a perfect conception of 

 proportion, though not perhaps of the manner of measuring magnitudes 

 with a view to express their proportions by means of numbers. Ail 

 who can trace the resemblance of a drawing to the original, or have 

 the least notion of the use of a map, are in possession of the funda- 

 mental notions on which a theory of proportion con be founded. The 

 term RATIO is that under which the first part of the subject should be 

 treated, and the article cited will contain matter preliminary to the 

 present one. It will be sufficient for our present purpose to state that 

 the ratio or relative magnitude of two magnitudes is to be measured 

 by the number of times or parts of times which one is cont 

 tho other, whenever the two ore commensurable ; and we shall now 

 confine ourselves to the purely mathematical treatment of the theory 

 of pro|x>rtion, and Bhall avoid, as much as the nature of the subject 

 will admit, all discussion of the notion of ratio considered as a 

 magnitude. 



.mm >t well explain the nature of the difficulty which occurs in 

 the theory of proportion, without a prior reference, first, to the purely 

 arithmetical treatment of the subject, and secondly, to the practical 

 sufficiency of this method which is the necessary consequence of our 

 physical constitution. 



If all magnitudes of the same kind were necessarily comment > 

 that is, if any on* among them being taken as a unit, the rest were all 

 expressible by multiples, aliquot parts or submultiples, and multiples 

 of aliquot parts, of the unit chosen, no difficulty would arise in making 

 the subject of proportion purely arithmetical. For let a and 6 rrpiv 

 sent the units, parts of unit*, or both, in two magnitudes of the same 

 kind (as two lines) ; any sufficient demonstration of tho nil. . 

 division of one fraction by another will show that a contains 6 pi 

 a number of times and parto of time*. If then we say that ci is to '; 

 in tho name proportion as c to d, we mean that o contains 6 precisely 



such times and ports of times u c contains if : that is, we assert tho 

 aquation 



l 



I - d' 



the mathematical treatment of which U so smy, that no one who con 

 solve a simple equation can be stopped for a moment by the difficulty 

 of any consequanoe of it. The following proposition, which may be 

 proved generally, contains all the consequences which an most useful. 

 Let a, o, c, and d be (in the arithmetical sense) proportional : take any 

 two functions of a and 6, which are homogeneous and of the same 

 dimension (such u o*+ 6* and a 3 o 1 ). Take corresponding functions 

 of e and ci (which are ed + d* and c 9 if 2 ) ; then the four numbers so 

 obtained are also proportional (that is, at + 6'' contains ' 4 s as many 

 times and port* of times ucd + d 1 contains <* ci-). 



In measuring magnitudes of which the numerical representatives 

 are afterwards to be submitted to calculation, it necessarily follows, 

 from the imperfections of our senses, that some imperceptible amount 

 of magnitude mutt always be neglected or added ; so that, for ex 

 that which we call a line of 8 inches long means something between 

 2-9 and 8-1, or 2-90 and 8'01, or 2-998 and 8-001, according to the 

 degree of accuracy of the measurement. All magnitudes therefore ore 

 practically commensurable; for suppose, in the case of weights for 

 example, that a grain is taken as the unit, and that the ten-thou- 

 part of a unit is considered as of no consequence ; then by taking 

 every weight only to the nearest ten-thousandth of a grain, they may 

 every one be expressed arithmetically with a conventional degree of 

 precision, which for every purpose of application will do as well as 

 though it were perfectly accurate. 



The discovery of INCOMMENSURABLE magnitudes, one of the most 

 striking triumphs of reason over the imperfection of the senses, was 

 mode at a very early period ; since the demonstration of their exist- 

 ence, the classification (to a certain extent) of their species [IRRATIONAL 

 QUANTITIES], and tho means of overcoming the difficulties which they 

 present, appear in the writings of Euclid. A moment's consideration 

 will show that a property of numbers, a relation of figure in geon- 

 a general law of nature, may be inferred from induction with a degree 

 of probability which will amount to moral certainty, both as to the 

 exactness and universality of the property, relation, or law. I'.ut the 

 existence of incommensurable magnitudes can never be made certain, 

 except by absolute deduction : no attempt at measurement, a minimum 

 viiibiU existing, could show the non-existence of any common measure, 

 however email. Suppose for instance that, having provided mei 

 measurement which can always be depended on to show the thousandth 

 of an inch, but nothing less, a person should accurately (as tho word 

 is commonly used) lay down squares of one, two, 4c., inches in tho 

 side, with a view to render the existence of a common measure to the 

 side and diagonal exceedingly improbable by experiment. If not 

 before, he would lie battled by the square whose side is 237S inches, 

 i In- diagonal of which could not by his measures be distinguished from 

 3363 inches, from which it differs only by about the fitt-tkoiuandtk 

 part of on inch. And let any greater degree of exactness be attained 

 in the means of measurement, short of positive accuracy, a reasoner on 

 the subject could still predict a square which should defeat the attain- 

 ment of the object sought. 



The mere existence of incommensurables, to say nothing of their 

 frequent occurrence, and the impossibility of avoiding them, i < 

 tho arithmetical theory of proportion inexact in its very definition. If 

 we would say, for instance, that the diagonal of a larger square is to its 

 side as the diagonal of a smaller square is to its side, we enunciate a 

 proposition the meaning of which is unsettled. For if we mean 

 to assert that tho larger diagonal contains the larger side as many 

 times or fractions of times as tho smaller diagonal contains the 

 smaller, it is answered, by those who wish for precise u< 

 that neither diagonal contains its side any exact number of time* 

 or parta of times. If we should say that the larger diagonal lies 

 between 1-414:213 and V414214 times tho larger side, and that 

 the smaller diagonal also lies between 1-414213 and I'll 1-1 i 

 the smaller side, and if we should show this to be true, we certainly 

 show that wo could produce lines very nearly equal to tho diagonals, 

 which arc, under the arithmetical definition, pr.'poitional to the sides; 

 and that this might be done without altering either diagonal by so 

 much as the millionth part of the side. And the ten-millionth, hundred- 

 inilliouth, or any aliquot part of the side, however small, might be 

 substituted for the millionth, without detriment to our power of 

 showing the truth of the proposition. If we use the means by which 

 this process may be carried on ad injinitum, we may perhaps be said to 

 have established the truth of the proposition that the diagon 

 squares are as their sides. But if we in any manner stop short of this, 

 we destroy the rigorous character of geometry, and produce a system 

 of mathematics which, like a common table of logarithms, is true to a 

 certain number of places of decimals, and not farther. It is ol 

 that such a system of mathematics, like the table with which v- 

 compared it, is sufficient for the purposes of practical applii -.ition : nor 

 have we the least quarrel with those who, desiring an iuHtruiucnt only, 

 .-. ith one which is sharp enough for their purpose. \\V 

 only complain of them when they assert and teach others that their 

 tool baa an edge keen enough to separate the minutest truth from the 



